Type problem and the first eigenvalue

Abstract

In this paper, we study the relationship between the type problem and the asymptotic behavior of the first eigenvalues ${\lambda }_1(B_r)$ of ``balls'' $B_r:=\{\rho <r\}$ on a complete Riemannian manfold $M$ as $r\to +\infty$, where $\rho$ is a Lipschitz continuous exhaustion function with $|\nabla \rho |\le 1$ a.e. on $M$. We show that $M$ is hyperbolic whenever \[ Λ_*:= \liminf_{r\rightarrow +\infty} \{ r^2 λ_1(B_r)\} >18.624\cdots. \] Moreover, an upper bound of ${\Lambda }_{\ast }$ in terms of volume growth ${\nu }_{\ast }:={\mathrm{liminf}}_{r\to +\infty }\frac{\log |B_r|}{\log r}$ is given as follows \[ {Λ_*} \lesssim \begin{cases} ν_*^2,\ \ \ &ν_*\gg1,\\ ν_*\log\frac{1}{ν_*},&1<ν_*\ll1. \end{cases} \] The exponent $2$ for ${\nu }_{\ast }\gg 1$ turns out to be the best possible.

21 pages. Comments welcome!